In practice, the actual probability distribution \(P\)  of \(\mbox{$\xi $}\)  is rarely known and we are only in possession of an approximating probability distribution obtained through modeling assumptions and data analysis. It is therefore critical to examine the effect of replacing \(P\)  by an alternative on the solutions of the corresponding optimization problems. With this goal in mind, we consider a sequence of probability distributions \(\lbrace P^\nu , \nu \in \mathbb {N}\rbrace \) , where \(\mathbb {N}= \lbrace 1, 2, 3, \dots \rbrace \) , tending to \(P\)  in some sense as \(\nu \rightarrow \infty \) . The sequence of distributions may not be the outcome of a specific estimation procedure and could be rather conceptual as part of a sensitivity analysis or outlier analysis; the study of influence functions in statistical learning [1]}, [2]} furnishes examples. In particular, \(P^\nu \)  may not be an empirical distribution using \(\nu \)  sample points, although this is a possibility within our framework. Regardless of the setting, one would hope that \(P^\nu \)  tending to \(P\)  ensures that optimization problems involving \(P^\nu \)  indeed produce arbitrarily accurate solutions of the actual problem defined by \(P\) . It is well known that caution is advised before jumping to such a conclusion.
